9000034709
Level:
A
Consider equation
\[
p(2 - p)x = 4p
\]
with a real parameter \(p\).
Solve the equation for \(p = 2\).
\(\emptyset \)
\(\mathbb{R}\)
\(\left \{ \frac{4}
{2-p}\right \}\)
\(\mathbb{R}\setminus \left \{0\right \}\)
Equations and Inequalities with Parameters
9000034702
Level:
B
Identify the set of the values of the real parameter \(d\) for which the following equation has no solution in \(\mathbb{R}\).
\[
x^{2} - 2dx + 2d^{2} - 9 = 0
\]
\((-\infty ,-3)\cup (3,\infty )\)
\((-3,3)\)
\((3,\infty )\)
\((-\infty ,-3)\)
9000034703
Level:
B
Identify the set of the values of the real parameter \(t\) for which the following equation has two different real roots.
\[
x^{2} + (t + 2)x + 1 = 0
\]
\((-\infty ,-4)\cup (0,\infty )\)
\((-\infty ,-4)\)
\((-4,0)\)
\((0,\infty )\)
9000034704
Level:
B
Solve the inequality
\[
ax - 2 > 0
\]
with a real unknown \(x\) and a
nonpositive real parameter \(a < 0\).
\(\left (-\infty , \frac{2}
{a}\right )\)
\(\left (-\infty ,-\frac{2}
{a}\right )\)
\(\left (\frac{2}
{a},\infty \right )\)
\(\left (-\frac{2}
{a},\infty \right )\)
9000034705
Level:
B
Solve the inequality
\[
2x + b > 0
\]
with a real unknown \(x\)
and a real parameter \(b\in \mathbb{R}\).
\(\left (-\frac{b}
{2},\infty \right )\)
\(\left (\frac{b}
{2},\infty \right )\)
\(\left (-\infty , \frac{b}
{2}\right )\)
\(\left (-\infty ,-\frac{b}
{2}\right )\)
9000034706
Level:
A
Consider the inequality
\[
px^{2} - 2x + 2 > 0
\]
with the real parameter \(p\).
Solve this inequality for \(p = 0\).
\((-\infty ,1)\)
\((-\infty ,-1)\)
\((-1,\infty )\)
\((1,\infty )\)
9000033704
Level:
B
Find the values of real parameter \(p\)
which ensure that the following quadratic equation has solutions with nonzero
imaginary part.
\[
px^{2} + 4x - p + 5 = 0
\]
\(p\in \left (1,4\right )\)
\(p\in [ 1,4] \)
\(p\in \left (-\infty ,1\right )\cup \left (4,\infty \right )\)
\(p\in \left (-\infty ,1\right ] \cup \left [ 4,\infty \right )\)
9000021706
Level:
A
Find all the values of the parameter \(k\) so that the solution of the following equation is bigger than \(10\).
\[
3x - 18 = \frac{10x - 4k}
{2}
\]
\(k\in (19,\infty )\)
\(k\in \{9\}\)
\(k\in (-\infty ,1)\)
\(k\in (9,\infty )\)
9000021707
Level:
A
Find all the values of the parameter \(k\) so that the solution of the following equation is positive.
\[
2kx + k = 4x + 3
\]
\(k\in (2,3)\)
\(k > 0\)
\(k\in (3,\infty )\)
\(k\in (-\infty ,3)\)